Theorem bj-resssxp 38176

Description: A class is equal to its restriction to some class if and only if it is included in the Cartesian product of the restricting class with the universal class. More fundamental form of resssxp 6294 (or at least, more entitled than it to that label). Can shorten relssres 6027 through the equivalence ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝐵 × V)), which could itself be added to set.mm. (Contributed by BJ, 26-Oct-2026.)

Assertion

Ref	Expression
bj-resssxp	⊢ ((𝑅 ↾ 𝐴) = 𝑅 ↔ 𝑅 ⊆ (𝐴 × V))

Proof of Theorem bj-resssxp

Step	Hyp	Ref	Expression
1		df-res 5668	. . 3 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
2	1	eqeq1i 2768	. 2 ⊢ ((𝑅 ↾ 𝐴) = 𝑅 ↔ (𝑅 ∩ (𝐴 × V)) = 𝑅)
3		dfss2 4097	. 2 ⊢ (𝑅 ⊆ (𝐴 × V) ↔ (𝑅 ∩ (𝐴 × V)) = 𝑅)
4	2, 3	bitr4i 281	1 ⊢ ((𝑅 ↾ 𝐴) = 𝑅 ↔ 𝑅 ⊆ (𝐴 × V))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   × cxp 5654   ↾ cres 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-in 4094  df-res 5668

This theorem is used by:  bj-dfrel3  38180  relssresb  38188